Shock-induced superheating and melting curves of geophysically important minerals

Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

Shock-induced superheating and melting curves of geophysically important minerals Sheng-Nian Luo∗ , Thomas J. Ahrens Lindhurst Laboratory of Experimental Geophysics, Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA Received 5 February 2003; received in revised form 22 April 2003; accepted 23 April 2003

Abstract Shock-state temperature and sound-speed measurements on crystalline materials, demonstrate superheating-melting behavior distinct from equilibrium melting. Shocked solid can be superheated to the maximum temperature, Tc . At slightly higher pressure, Pc , shock melting occurs, and induces a lower shock temperature, Tc . The Hugoniot state, (Pc , Tc ), is inferred to lie + = Tc /Tc − 1. Shock-induced along the equilibrium melting curve. The amount of superheating achieved on Hugoniot is, ΘH superheating for a number of silicates, alkali halides and metals agrees closely with the predictions of a systematic framework describing superheating at various heating rates [Appl. Phys. Lett. 82 (12) (2003) 1836]. High-pressure melting curves are constructed by integration from (Pc , Tc ) based on the Lindemann law. We calculate the volume and entropy changes upon melting at (Pc , Tc ) assuming the R ln 2 rule (R is the gas constant) for the disordering entropy of melting [J. Chem. Phys. 19 (1951) 93; Sov. Phys. Usp. 117 (1975) 625; Poirier, J.P., 1991. Introduction to the Physics of the Earth’s Interior. Cambridge University Press, Cambridge, 102 pp.]. (Pc , Tc ) and the Lindemann melting curves are in excellent accord with diamond-anvil cell (DAC) results for NaCl, KBr and stishovite. But significant discrepancies exist for transition metals. If we extrapolate the DAC melting data [Phys. Rev. B 63 (2001) 132104] for transition metals (Fe, V, Mo, W and Ta) to 200–400 GPa where shock melting occurs, shock temperature + measurement and calculation would indicate ΘH ∼ 0.7–2.0. These large values of superheating are not consistent with the superheating systematics. The discrepancies could be reconciled by possible solid–solid phase transitions at high pressures. In particular, this work suggests that Fe undergoes a possible solid–solid phase transition at ∼200 GPa and melts at ∼270 GPa upon shock wave loading, and the melting temperature is ∼6300 K at 330 GPa. © 2004 Elsevier B.V. All rights reserved. Keywords: Superheating; Melting curve; Shock waves; Sound-speed; Molecular dynamics

1. Introduction Since the initial studies of dynamic compression of gabbro and dunite (Hughes and McQueen, ∗ Corresponding author. Present address: P-24 Plasma Physics, Los Alamos National Lab, MS E526, Los Alamos, NM 87545, USA. Tel.: +1-505-664-0037; fax: +1-505-665-3552. E-mail address: [email protected] (S.-N. Luo).

1958), it has been recognized that polymorphic, melting and freezing transitions in minerals often require excess pressure and temperature relative to the equilibrium phase boundary. In practice hysteresis has long been observed in many phase transformations (Taylor et al., 1991). Of particular interest is the superheating-melting behavior of shock-loaded crystalline materials. Shock melting experiments are useful for studying high-pressure melting of

0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2003.04.001

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geophysically and physically important materials. Shock-state sound-speed and temperature measurements with planar shock wave loading on silicates, alkali halides and metals have demonstrated the existence of superheating-melting behavior which is distinct from equilibrium melting. Superheating concept has been applied to interpret the sharp drops in temperature along Hugoniots (Lyzenga et al., 1983; Boness and Brown, 1993; Holland and Ahrens, 1997; Boness, 1999). Superheating has also been observed in intense laser irradiation (Williamson et al., 1984; Fabricius et al., 1986; Herman and Elsayed-Ali, 1992; Murphy et al., 1993). We attempt here to reconcile the issues related to the interpretation of temperature measurements in shock melting experiments, and whether the large discrepancies between the melting temperatures obtained from shock wave loading and recent diamond-anvil cell (DAC) (Errandonea et al., 2001) results for transition metals such as Fe, V, Mo, W, and Ta, can be simply attributed to the effect of superheating. Of special geophysical interest is the discrepancy between the shock melting data (Ahrens et al., 2002) and extrapolated DAC melting curve (Boehler, 1993) for iron. Previously, there were no systematic theoretical investigations (either phenomenological or firstprinciples) directly addressing shock-induced superheating. The lack of a practical superheating theory is partly due to the difficulty in achieving superheating in low heating-rate experiments, where heterogeneous nucleation dominates. Heterogeneous nucleation is favored at free surfaces, grain boundaries and impurities, etc., which significantly lower the energy barriers for nucleation. Homogeneous nucleation of crystals by undercooling a liquid is more manageable experimentally and a body of data have been documented (Kelton, 1991), while appreciable superheating at low heating rates has been observed in very few cases using special experimental designs (Uhlmann, 1980). Ultrafast heating (109 to 1013 K/s) is characteristic of dynamic melting experiments and is responsible for the observed superheating, as the temperature can rise faster than the rate of rearrangement of atoms required for melting. To quantify the amount of superheating and relate superheating to melting at high pressures, it is important to predict the maximum superheating achievable at heating rates comparable to those in shock wave loading. Previous theories prof-

fered involve the catastrophes of entropy (Fecht and Johnson, 1988), rigidity and volume (Tallon, 1989) at melting. Other efforts to describe superheating utilized kinetic nucleation theory (Lu and Li, 1998; Rethfeld et al., 2002). These theories are not applicable because they either predict too high an upper bound to superheating or do not take heating rate into account. Superheating is internally limited by the nucleation energy barrier and externally by the heating process. Recently we developed a framework of systematics to relate superheating to undercooling parameters, systematics of energy barriers for nucleation, and heating rates (Luo and Ahrens, 2003; Luo et al., 2003). In this paper we review our theory, survey the existing shock compression superheating data and discuss their implications to high-pressure melting of silicates, alkali halides and transition metals.

2. Superheating systematics Our previous work on superheating and undercooling systematics (Luo and Ahrens, 2003; Luo et al., 2003) is based on homogeneous nucleation theory, undercooling experiments, and molecular dynamics simulations of melting and refreezing. 2.1. Homogeneous nucleation theory and systematics of energy barrier for nucleation The nucleation rate per unit volume, I, for steadystate homogeneous nucleation, can be written as (Turnbull and Fisher, 1949; Christian, 1965; Walton, 1969; Motorin and Musher, 1984; Kelton, 1991)

 Ea Gc I = g(m, T) exp − exp − (1) kT kT where g(m, T) is a function of material properties (m) and temperature (T ), Ea the activation energy, Gc the critical Gibbs free energy for nucleation, and k the Boltzmann constant. Or, more simply (Rethfeld et al., 2002; Kelton, 1991)
 Gc I = M(m, T) exp − (2) kT where M is a function of material properties m, such as the heat of fusion per unit volume Hm , melting temperature Tm , solid–liquid interfacial energy

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

γsl , viscosity, heat capacity and critical size, etc. The T -dependence of M(m, T) is negligible compared to the exponential Gc term, thus M(m, T) can be regarded as a constant, I0 (Walton, 1969; Porter, 1981; Kelton, 1991). Consider a spherical liquid nucleus of critical radius rc (the minimum size beyond which the energy of formation of nuclei decreases) inside a perfect crystalline lattice, the critical nucleation energy Gc (Christian, 1965) is Gc =

16πγsl3 3(Gsl − E )2

371

We define the energy barrier for nucleation, β, as a dimensionless quantity (Luo and Ahrens, 2003) β(γsl , Hm , Tm ) =

16πγsl3 3Hm2 kTm

(6)

and introduce reduced temperature, θ = T/Tm . Thus I = I0 f(β, θ) with   β f(β, θ) = exp − (7) θ(θ − 1)2 The prefactor I0 can be obtained experimentally or theoretically. Nucleation is essentially controlled by f(β, θ), namely, by the dimensionless quantity β at given temperature. The form of f(β, θ) is simple, but it reflects the physics of nucleation. As shown in Fig. 1, during superheating of solids, f (i.e., I/I0 ) increases with temperature monotonically, as the mobility of atoms and the chemical driving force for melt nucleation both increase with T . Upon undercooling of liquids, the chemical driving force induced by undercooling is partly offset by the decrease in mobility, thus there is a maximum for f at θ = 1/3. Although diffusion processes are not explicitly included in f , they were accounted for, by the functional form of f . The dimensionless quantity β is characteristic of a material and depends on γsl , Hm and Tm . An estimate the value of β can be obtained from hard sphere

(3)

where Gsl is the Gibbs free energy difference per unit volume between solid and liquid state, approximated as
 T (4) − 1 Hm Gsl ≈ Tm assuming that the heat capacities of solid and liquid are equal (Turnbull, 1952; Fecht and Johnson, 1988; Kelton, 1991). E is the strain energy due to the volume change between solid and liquid. We assume E = 0 (the effect of E is small). I is often reduced to the form (Walton, 1969; Porter, 1981; Kelton, 1991)   16πγsl3 Tm2 (5) I = I0 exp − 3Hm2 kT(T − Tm )2

1.0 UC SH

0.8

0.01

0.1

f (β , θ )

1

0.6

0.01

5 10

0.4 0.1

0.2 θc

0.0 0

1

2

3

θ = T / Tm Fig. 1. f(β, θ) vs. reduced temperature θ for different values of β. Each curve is labeled with the adopted value of β. UC denotes undercooling and SH superheating. θc is the critical temperature obtained from the tangent method.

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Table 1 Normalized energy barrier β for elements, alkali halides and silicates at ambient pressure (Luo et al., 2003) Ag Au Cd Co Cu Fe Hf Hg Mn Mo

1.67 1.48 1.64 1.69 1.51 3.11 1.62 6.30 2.12 1.31

Ni Pd Pt Rh Ru Ta Ti V W Zr

2.63 1.66 1.73 1.99 0.86 1.96 1.45 1.95 1.55 2.00

Al Bi Ga Ge In Pb Sb Se Sn Te

1.47 5.83 8.15 4.87 3.04 5.87 2.48 0.20 5.34 4.43

systems (hss) which are simple and yet have theoretical significance. Let the diameter of a hard sphere be σ. The solid–liquid interfacial energy for a hard sphere (γslhss ), is shown to be 0.61kT/σ 2 (Davidchack and Laird, 2000). Heat of fusion per unit volume for hss, is Hmhss = 1.16kT/V hss , where V hss = πσ 3 /6 (Hoover and Ree, 1968). Thus, β ∼ 0.77 for a hss. We may expect for a real system, the value of β is of a similar magnitude. For materials of interest, γsl ∼ 0.1 J/m2 , Tm ∼ 103 K and Hm ∼ 109 J/m3 (Kelton, 1991) yields β ∼ 1.2. The elements studied in undercooling experiments (Kelton, 1991; Shao and Spaepen, 1996) include transition metals, simple metals and semiconductors in the third to sixth row of the periodic table.

LiF LiCl LiBr NaF NaCl NaBr KCl KBr KI RbCl

2.05 2.05 0.63 2.22 1.27 1.27 1.27 1.27 1.27 1.27

Ga Hg

Energy Barrier (β )

8

Pb Bi

6

Sn Ge

Te

4 Fe

In Ni

V

2

Zr Co

Al

Ta

Mn

Ti

Cu

Pt

Rh

Sb Cd Pd Ag

Mo

Hf W

Au

Ru

Se

0 10

20

30

1.27 1.41 1.56 2.40 1.52 5.01

The systematics for elements may be valid for materials in general. The values of β for these elements are calculated (Table 1 and Fig. 2). Transition metals (Group IVB–IIB) have relatively constant value of β with an average close to 2, except for Hg (β = 6.30). Group IIA–VIA elements have significantly higher values of β, with maximum at Ga (β = 8.15) except for Al (β = 1.47) and Se which has the lowest β = 0.20. These variations largely reflect the periodicity of electronic structure of elements. As γsl increases with Tm and Hm (Luo et al., 2003), the variations in β are relatively small from material to material. It is demonstrated that the quantity γg /Hm V is nearly a constant for a variety of elements and compounds

10

Si

CsF CsCl CsBr CsI Albite Quartz

40

50

70

80

Atomic Number (Z) Fig. 2. Systematics of β for elements (β vs. Z) (Luo and Ahrens, 2003).

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373

(Kelton, 1991; Uhlmann, 1980). The gram-interfacial 1/3 energy is γg = γsl V 2/3 NA , where V is the molar volume and NA the Avogadro’s number. As Hm = Tm Sm , where Sm is the entropy change upon melting, thus from its definition β is proportional to Sm V , i.e., to the entropy of melting per mole atoms (∼R). Therefore β should vary over a narrow range for different materials. The same result can be obtained from the argument that γsl ∝ Hm V/a2 , where a2 is the effective atomic surface area (Walton, 1969). As the entropy of melting (per mole atoms) at high pressures is not significantly different from that at ambient pressure (see Section 4), we expect that β for most solids under high pressures (e.g., megabar pressure under shock compression) should be of the same order as at ambient pressure. Moreover, molecular dynamics simulations of superheating and undercooling of Al at 0–100 GPa demonstrate a weak pressure-dependence of β (Luo et al., 2003). Values of β for alkali halides and silicates are also listed in Table 1.

heating (or undercooling). This method (referred to as tangent method) is easy to visualize, but its difficulty lies in the arbitrariness of choosing f0 which is not defined from first-principles yet. Given the β systematics, next we develop a Q-dependent (Q is the heating or cooling rate) scheme to probe the systematics of superheating characteristic of elements and compounds. The undercooling experiments on various elements and compounds were documented (Kelton, 1991), and serve the basis for our calculating the superheating at various heating rates. The parameters for undercooling such as γsl , Hm , Tm and molar volume V can be regarded as equal to those for superheating. In the case of superheating, the probability (Kelton, 1991) x of v moles of parent phase containing no new phase (liquid) under heating rate Q+ is    + vTm I0 V θc x = exp − f(β, θ) dθ (9) Q+ 1

2.2. The maximum amount of superheating

and the expression is similar for the undercooling case. If we assume I0 , v, and x for superheating are similar to those for undercooling, the maximum superheating temperature θc+ can be estimated from the documented θc− using Eq. (9). Given undercooling θc− (Kelton, 1991) under typical cooling rate Q− = 1 K/s, θc+ under certain heating rate Q+ can be obtained by solving  1  θc+ 1 1 f(β, θ) dθ = f(β, θ) dθ (10) Q− θc− Q+ 1

The limit of superheating was investigated by assuming an absolute value of I = 1 cm−3 s−1 (Lu and Li, 1998) for some metals. Similarly, assuming the nucleation time tM = (IVc )−1 , where Vc = (4/3)πrc3 , the critical volume of a nucleus with critical radius rc , the maximum superheating can be estimated at given nucleation time scale (referred to as the Vc method) (Rethfeld et al., 2002). Consider the normalized nucleation rate, I/I0 = f(θ, β) (Fig. 1). For nucleation of melt in a superheated solid, virtually no nuclei are formed until a critical temperature Tc (or θc = Tc /Tm ) is reached at which point a catastrophic nucleation occurs (Porter, 1981). It is similar for nucleation of crystal in an undercooled liquid. θc+ or Θ+ = θc+ − 1 refers to the maximum superheating, and θc− or Θ− = 1 − θc− to the maximum undercooling. (Superscript + denotes superheating and − undercooling.) First we construct a tangent of f(β, θ) at a certain f for a given β, say, f0 = 0.1 (the dashed line in Fig. 1). The tangent line with a slope  ∂f  (8) ∂θ  θ=θ(f0 ,β)

intersects the θ axis at θc (dotted line in Fig. 1). This then yields a simple treatment of the maximum super-

For superheating, we consider heating rates between Q+ = 1 K/s (corresponding superheating denoted as + ). These heating rates should Θs+ ) and 1012 K/s (Θns be regarded as typical but not exact, because a factor of 102 difference in Q would yield a negligible difference in θ given a reasonable value of f(β, θ). θc+ at typical heating rates is calculated as shown in Fig. 3 for elements along with experimental values for θc− . The numerical relationship between β and θc at given Q can be fitted with a simple analytical form (Luo et al., 2003) β = (A0 − b log Q)θc (θc − 1)2

(11)

where A0 = 59.39, b = 2.33 and Q is normalized by 1 K/s. This functional form applies to both superheating and undercooling. Thus the material property β,

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10

undercooling superheating Q = 1 K/s

Ga

Q = 1 K/s

8

Hg Bi,Pb Sn

6

β

Q = 106 K/s

Ge Te

4 In,Fe Ni Sb

Q = 1012 K/s

Q = 106 K/s

2 Ru Se

0

0

0.5

1.0

Q = 1012 K/s

1.5

θc = Tc / Tm Fig. 3. Maximum undercooling and superheating (β vs. θc ) of elements under various heating (cooling) rates Q. θc in superheating regime is calculated with Eq. (10). Note that there are a maximum for β and a minimum for θc (maximum undercooling) in the undercooling regime for each Q. Both superheating and undercooling under certain Q are described in Eq. (11). Solid curves are plotted with different values of Q, and dotted curves denote the non-physical portion of the undercooling. The elements within the double-headed arrow are Ti, Al, Au, Cu, Hf, Cd, Pd, Ag, Co, Pt, Ta, Rh, Zr, Mn, Si, Sb, Ni, In and Fe in β-increasing order. After Luo et al. (2003).

the maximum superheating (undercooling), and heating (cooling) process are related via Eq. (11). The similarity between Eqs. (7) and (11) simply indicates a catastrophic behavior of nucleation near θc . The limit for superheating Θ+ is inherently limited by the material property β, and increases monotonically with β and Q (Fig. 3). In general, the energy barrier β is in the range of 0.2–8.2 (Table 1). For Q = 1 to 1012 K/s, Θ+ varies in a narrow range of 0.05–0.43 for all elements considered, with maximum at Ga and minimum at Se. A change in heating rate from 1 to 1012 K/s induces an increase of less than 10% in the maximum superheating (θc ). Thus, the effect of Q is more important in achieving superheating experimentally (Section 3) than achieving larger superheating. Group IIIA–VIA elements (except Se) can be superheated more than transition elements (except Hg). Transition metals such as Fe, V, Ta, Mo and W can be superheated by 0.30Tm at most (we expect that β at high pressures, does not vary much as argued before). At similar heating and cooling rates, the maximum undercooling (Θ− ) is larger than maximum superheating (Θ+ ).

A partial verification of the above superheating systematics can be achieved through molecular dynamics simulations. The heating (and cooling) rate in typical MD simulations is ∼1012 K/s. For a perfect supercell with three-dimensional periodic boundaries, superheating has long been observed in MD simulations of melting (Allen and Tildesley, 1987). There are few systematic studies addressing superheating previous to Luo et al. (2003). MD simulations were conducted for f.c.c. metals and Be (h.c.p.) with quantum-corrected Sutton–Chen many-body potentials and single- and two-phase simulation techniques (Luo et al., 2003). The crystal melts at superheated temperature T1,m in single-phase simulations, and the equilibrium melting temperature is obtained from solid–liquid coexisting phase (two-phase) simulations as T2,m . Thus, superheating achieved in MD is + Θmd = T1,m /T2,m − 1, and listed in Table 2. The sim+ ) of the ulations agree closely with the prediction (Θns superheating systematics. It is also worth mentioning that both superheating and undercooling achieved in these MD simulations (Luo et al., 2003), are in accord with the superheating–undercooling systematics

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

Be Al Ni Cu Rh Pd Ag Ir Pt Au Pb Ta

exp

T1,m (K)

T2,m (K)

Tm

1600 1100 1700 1350 2700 1850 1200 3400 2450 1400 700 3650

1350 925 1375 1070 2125 1475 1000 2740 1925 1075 575 3150

1560 933 1728 1356 2239 1825 1234 2683 2042 1336 601 3253

(K)

+ Θmd

+ Θns

0.19 0.19 0.24 0.21 0.27 0.25 0.20 0.24 0.27 0.30 0.22 0.15

– 0.20 0.26 0.19 0.23 0.21 0.21 – 0.21 0.20 0.37 0.23

exp

Tm is the experimental value of Tm . Ta (b.c.c.) results are from Strachan et al. (2001).

(Eq. (11)). In MD simulations of high-pressure melting for stishovite, -Fe and the Lennard-Jones system with different potentials, similar values of maximum superheating were observed (Belonoshko et al., 2000; Luo et al., 2002a, 2003, 2004a). Given the above methods of estimating maximum superheating, we compare their predictions. The melting theories based on catastrophe of entropy (Fecht and Johnson, 1988), catastrope of rigidity and volume (Tallon, 1989) predict a wide range of superheating of Θ+ = 0.2–2.0 for crystalline solids (Lu and Li, 1998). In particular for Al, Θ+ is 0.38, 0.29 and 0.24, for entropy, rigidity and volume catastrophe respectively, while by assuming I = 1 cm−3 s−1 and homogeneous nucleation, Θ+ = 0.21 (Lu and Li, 1998). For Al (β = 1.47), the tangent method predicts Θ+ = 0.51 with f0 = 0.1 and Θ+ = 0.16 with f0 = 10−20 . Note that these methods do not involve heating rates. The Vc method predicts Θ+ = 0.34 for Al with tM = 1 ps, and the superheating systematics predict Θ+ = 0.16 and 0.23 at Q = 1 and 1012 K/s, respectively. The present systematics based on undercooling experiments and β-systematics define tightest upper bounds for superheating and incorporate effects of heating rates.

3. Superheating achieved in ultrafast experiments Both diamond-anvil cell experiments and shock wave loading can be useful tools to study the melting

behavior under extreme conditions. The temperature achievable and measurable in high-pressure DAC melting experiments, is limited by laser-heating techniques. Heating in DAC experiments is essentially steady-state. In contrast, pressure and temperature are raised simultaneously on ultrashort time scales upon shock compression. The typical risetime of a strong shock front for planar impact is of the order of nanosecond (ns), and the temperature increase is ∼103 K, thus the effective heating rates Q is ∼1012 K/s. Q is similar in intense laser irradiation experiments, depending on energy deposited, irradiation time and material properties. We note that in shock wave experiments, solids can be heated at rates higher than those required for rearrangement of atoms upon melting. During shock wave loading, solids are heated inside as the shock front advances, thus surface melting can be suppressed. Kinetics inherent in melting may play an important role at these time scales, thus inducing superheating. Fig. 4 shows the schematic diagram of the two distinct types of melting behavior upon shock wave loading: equilibrium and non-equilibrium melting. For equilibrium melting, the T − P Hugoniot states lie on

d liquid

c' superheated solid

MC

c

T

Table 2 Molecular dynamics simulations (Luo et al., 2003) of closed-packed metals (except Ta) with quantum-corrected Sutton–Chen potential at P = 0

375

b solid

a

P Fig. 4. Schematic of equilibrium and non-equilibrium melting upon shock wave loading. MC is the equilibrium melting curve and assumed to have positive dTm /dP slope. For equilibrium melting, the T − P Hugoniot states lie on abcd, where bc coincides with MC. The Hugoniot represented by abc cd indicates that shocked solid is superheated above MC. As P increases, T in successive Hugoniot states reaches a maximum (Tc ) and then drops to Tc upon melting near Pc . The maximum superheating on the Hugoniot + = Tc /Tc − 1. with non-equilibrium melting, is ΘH

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6000

NaCl Hugoniot point Kormer, 1965

5000

Ahrens et al., 1982 Boness, 1991

liquid

4000

T (K)

c' solid

c

3000

2000

Lindemann MC

B1 B2 DAC melting point Boehler et al., 1997

1000

0

20

40

60

80

100

P (GPa) Fig. 5. Shock-state temperature and DAC melting point measurements for NaCl.

order within the shocked crystal (Williamson et al., 1984). Shock melting experiments have been conducted on alkali halides (NaCl, CsBr and KBr), silicates (fused 5.4

Sound speed (km/s)

CsBr Boness & Brown, 1993

5.2

5.0

liquid solid

4.8

4.6 6000

5000

T (K)

abcd where bc coincides with the equilibrium melting curve. The Hugoniot represented by abc cd indicates that shocked solid is superheated above the melting curve. As P in successive Hugoniot states increases, T reaches a maximum (Tc ) and then drops to Tc upon melting near Pc . The maximum superheating on + the Hugoniot with non-equilibrium melting, is ΘH = Tc /Tc − 1. The reason for the preconception of equilibrium melting on Hugoniot is two-fold: equilibrium thermodynamics was assumed previously for convenience, and more importantly, the pioneering shock temperature measurements on NaCl (Kormer, 1965) demonstrate sharp T − P slope changes between 50 and 65 GPa (Fig. 5), thus appeared to lend experimental support to the equilibrium interpretation. As shown in the following analysis of static and dynamic melting experiments, superheating-melting upon shock loading is a dominant feature in shock melting experiments of non-porous materials. The techniques employed to detect shock melting are sound-speed and temperature measurements at the shock states. Upon melting, the sound-speed drops from longitudinal to bulk sound-speed due to the loss of rigidity. Similarly, the shock temperature in successive Hugoniot states drops as the solid melts due to latent heat (e.g., Fig. 6). Other techniques such as transient electron diffraction are important diagnostics to detect melting from the loss of long-range

solid

c'

liquid Lindemann MC

c

4000

3000 Boness & Brown, 1993

2000

0

20

40

60

80

100

P (GPa) Fig. 6. Shock-state sound-speed and temperature measurements for CsBr. The thin dashed line denotes the onset of melting.

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386 6.5

Sound speed (km/s)

KBr

6.0 solid liquid

5.5

Boness & Brown, 1993

5.0 Hugoniot point Boness & Brown, 1993

6000

liquid

5000 solid

T (K)

4000

c' c

3000

Lindemann MC

2000

377

we interpret the shock wave data (Ahrens et al., 1982; Kormer, 1965) as non-equilibrium melting (Fig. 5), and the estimated superheating agrees with the superheating systematics. The constructed Lindemann curve also agrees exactly with the DAC results. Thus, by considering superheating, we in principle can obtain one point, (Pc , Tc ), on the melting curve. Similar features have been observed in shock-state sound-speed and/or temperature measurements on other materials. These measurements include soundspeed and temperature measurement on fused quartz, quartz (Lyzenga et al., 1983; McQueen, 1992) (Figs. 8a and 9), Fe (Brown and McQueen, 1982, 1986; Williams et al., 1987; Yoo et al., 1993) (Fig. 10) and V (Dai et al., 2001) (Fig. 11), temperature measurement on forsterite (Holland and Ahrens, 1997) (Fig. 8b),

DAC melting point Boehler et al., 1996

c'

(a) SiO2 0

20

40

liquid

P (GPa)

c'

Fig. 7. Shock-state sound-speed and temperature, and DAC melting point measurements for KBr. The thin dashed line denotes the onset of melting.

solid

5000

c

T (K)

solid

c 4000 FQ

Lindemann MC

Q

DAC melting point Shen & Lazor, 1995 Hugoniot point Lyzenga et al., 1983

3000

2000 0

50

100

150

P (GPa) 8000 (b) Mg2SiO4 6000

c' solid

liquid

T (K)

quartz, quartz and forsterite), and transition metals (Fe, V, Mo, Ta). Alkali halides are advantageous for studying melting behavior due to their optical transparency and relatively low melting temperatures upon shock wave loading. The P − T conditions on Hugoniots are accessible in DAC apparatus. Simultaneous measurements of shock-state sound-speed and temperature for CsBr and KBr (Boness, 1991; Boness and Brown, 1993) are shown in Figs. 6 and 7. The simultaneous drop in sound-speed and temperature at ∼38 GPa (CsBr) and ∼29 GPa (KBr) signals melting. The amount of superheating is estimated to be 0.16 and 0.20 for CsBr and KBr, respectively. If we assume the energy barrier β of the high-pressure B2 phases of CsBr and KBr to be the same as ambient values, the superheating systematics predict superheating + ) of 0.20 and 0.18 respectively with Q = 1012 K/s (Θns (Table 3). (Pc , Tc ) is assumed to locate on the equilibrium melting curve, and it coincides with the DAC results (Boehler et al., 1996) for KBr. We also construct a Lindemann melting curve (discussed in next section) integrated from (Pc , Tc ). The static data agree almost exactly with the Lindemann curve. For NaCl,

liquid

6000

60

c

4000

schematic MC

Holland & Ahrens, 1997

2000 50

100

150

200

P (GPa)

Fig. 8. Shock-state temperature measurements for fused quartz and quartz (a) and forsterite (b). The DAC measurement on silica melting curve is also shown in (a). The Lindemann MC is constructed from (Pc , Tc ) on the principal Hugoniot of fused quartz (FQ), and that from quartz Hugoniot (Q) is about 150 K lower.

378

Starting phase

NaCl (B1) KBr (B1) CsBr (B2) SiO2 (fused) SiO2 (quartz) Fe (b.c.c.) V (b.c.c.)

Ending phase B2 B2 B2 Rutile Rutile Unknown Unknown

Superheating + ΘH

+ Θns

0.20 0.20 0.16 0.17 0.30 0.25 0.24

0.18 0.18 0.20 − − 0.23 0.26

Tc (K)

Pc (GPa)

ρ0 (g/cm3 )

sa

C0 a (km/s)

γ0 a

qb

dTm /dP c (K/GPa)

V/V c (%)

S(R)c

3125 3500 4260 4500 4800 5800 6150

57.8 33.0 43.0 70.0 113.0 270.0 220.0

2.232 3.071 4.440 4.310 4.310 8.280f 6.110

1.22 1.20 1.25 1.08d 1.24d 1.47f 1.23h

3.62 3.14 2.25 12.25d 9.66d 4.64f 5.06h

0.77 0.75 0.81 1.35e 1.35e 1.70g 2.00i

1.0 1.0 1.0 2.6e 2.6e 1.0g 1.0

10.88 21.92 22.09 10.87 9.33 8.53 14.98

0.85 1.02 0.91 1.58 1.47 1.19 1.83

0.704 0.708 0.705 0.744 0.730 0.732 0.761

Obtained from finite strain, linear US –up relation (Jeanloz and Grover, 1987) unless stated otherwise. Assumed to be 1 unless stated otherwise. c Evaluated at (P , T ). c c d Metastable Hugoniot centered on the high-pressure phase. e Luo et al. (2002b). f Brown and McQueen (1982). g Brown (2001). h Gathers et al. (1983). i Dai et al. (2001). a

b

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

Table 3 The maximum superheating achieved in shock wave loading, and the deduced melting parameters based on the Lindemann law and R ln 2 rule

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

SiO2

Sound speed (km/s)

16

solid

14 liquid

12 liquid

10 Qtz, McQueen, 1991 Qtz, Chhabildas & Miller, 1985 Fused Qtz, McQueen, 1991

8

40

60

80

100

120

140

P (GPa)

Fig. 9. Shock-state sound-speed measurements for fused quartz and quartz. The vertical dotted lines at ∼70 (on the principal Hugoniot of fused quartz) and 110 GPa (on the principal Hugoniot of quartz) indicate melting of stishovite (McQueen, 1992).

and sound-speed measurement on Mo (Hixson et al., 1989) (Fig. 12) and Ta (Shaner et al., 1984) (Fig. 13). The superheating achieved on these Hugoniots (except for forsterite) is in reasonable agreement with the predictions of the superheating systematics (Table 3). The superheating achieved on the principal Hugoniot + of forsterite is estimated to be ΘH = 0.63 which corresponds to β ∼ 20, an energy barrier exceptionally higher than those for any other material tested so far. Although we argued that β does not vary significantly at high pressures, we cannot rule out the possibility of β ∼ 20 for forsterite (or new phases from decomposition) under shock compression. Theoretical efforts (e.g., MD simulations) in resolving variations of β (essentially γsl ) with pressure, are of immediate necessity. Other causes could also well account for such a large superheating, such as underestimated emissivity (Luo et al., 2004b). Intense laser irradiation experiments have been conducted on Al (Williamson et al., 1984), Pb(1 1 1) (Herman and Elsayed-Ali, 1992), Bi(1 0 0 0) (Murphy et al., 1993) and GaAs (Fabricius et al., 1986), and the superheating achieved compares favorably to the present predictions of the superheating systematics. The shock melting experiments demonstrate a common feature of superheating with sharp drops in achieved shock temperatures, a behavior deviating from equilibrium melting. For complete melting of the solid (in the shock front) near Pc , the drop in T

379

relative to Tm due to the latent heat, can be approximated to the first order as Sm /Cp ∼ 1/3. Here Sm is the heat of fusion per mole atoms and assumed to be ∼R, and Cp is ∼3R. This temperature excess is comparable to the predictions of the superheating systematics, and the superheating achieved along Hugoniot assuming that (Pc , Tc ) lies along the equilibrium melting curve. The amount of maximum superheating achieved is sufficient not only to initiate the melting at certain Q, but also to compensate latent heat for complete melting, although these two processes are different physically. Thus, such an interpretation of shock melting experiments is reasonable, and an important point on the high-pressure melting curve, (Pc , Tc ) can be resolved. 4. Melting parameters induced from shock wave experiments Given (Pc , Tc ) from shock melting experiments, next we deduce other important thermodynamic parameters of melting, such as the melting curve, changes in entropy and volume upon melting at high pressures. In the absence of first-principles theories, the Lindemann law and the R ln 2 rule (systematics of the entropy of melting) are adopted. 4.1. The R ln 2 rule for entropy of melting First we briefly examine the validity of the R ln 2 rule. Previously, efforts have been made to establish the systematics of the entropy of melting (Oriani, 1951; Stishov, 1975; Tallon and Robinson, 1982; Poirier, 1991). The entropy of melting per mole atoms, Sm , can be written as Sm = SD + SV

(12)

where SV is the entropy of fusion associated with volume and vibrational changes, and SD the entropy of disorder (Oriani, 1951). Later, a R ln 2 rule was proposed (Stishov, 1975; Tallon and Robinson, 1982) for simple atomic and some binary and ternary substances, as Sm = R ln 2 + αKVm

(13)

where R is the gas constant, α the thermal expansivity of volume, K the bulk modulus, and Vm the volume

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11

Sound speed (km/s)

Fe 10 liquid

9 solid

8 Brown & McQueen, 1982

7 10000 Williams et al., 1987 Yoo et al., 1993

8000

liquid solid

T (K)

Lindemann MC

6000

4000

2000

0

DAC MC (Boehler, 1993)

γ α ε

0

100

200

300

400

P (GPa) Fig. 10. Shock-state sound-speed and temperature, and DAC melting point measurements for Fe. The dotted curve is the speculated melting curve connecting -iron and a possible unknown solid phase.

change upon melting. Comparing Eqs. (12) and (13) yields that SD is essentially R ln 2, although Oriani (1951) used a different approach to quantify disordering upon melting. The R ln 2 rule was justified for atomic systems with a simple cell model (Stishov, 1975) and by calculating the topological entropy (Rivier and Duffy, 1982). The R ln 2 rule has been checked against limited data and the universality is not well established. We take the entropies for elements (Table 4) (Oriani,

1951) and plot Sm versus SV . As shown in Fig. 14, the R ln 2 rule applies as expected. Note that there are exceptions, e.g., Ga (not plotted). The behavior of silicates is of particular interest. For the five silicates investigated (Table 5) (Poirier, 1991), the data points of forsterite, fayalite, pyrope and enstatite cluster around the R ln 2 line, except quartz for which SD is essentially 0 (Fig. 14). An appreciable amount of data for alkali halides was collected in Table 6 (Jackson, 1977). Entropies for KX (X = F, Cl, Br and I) agree

Table 4 The R ln 2 rule: entropy of melting for metals (values of entropies are from Oriani (1951) and normalized by R)

Sm (R) SV (R) SD (R)

Ag

Al

Au

Cu

Pb

Li

Na

K

Cs

Fe

Cd

Mg

Sn

1.102 0.407 0.695

1.378 0.528 0.850

1.142 0.654 0.488

1.152 0.367 0.785

1.026 0.397 0.629

0.770 0.050 0.720

0.855 0.100 0.755

0.855 0.099 0.756

0.830 0.091 0.739

0.991 0.100 0.891

1.237 0.513 0.724

1.177 0.538 0.639

1.715 0.774 0.941

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

381

Table 5 The R ln 2 rule: entropy of melting per mole atoms for silicates (n is the number of atoms per formula) n (cm3 /g)

Silicate Quartz (SiO2 ) Forsterite (Mg2 SiO4 ) Fayalite (Fe2 SiO4 ) Pyrope (Mg3 Al2 Si3 O12 ) Enstatite (MgSiO3 )

Vm a (cm3 /g)

3 7 7 20 5

2.50 3.80 3.70 18.3 5.40

Vm /V (cm3 /g)

αb (×10−6 K−1 )

K

0.105 0.081 0.076 0.158 0.160

24.3 30.6 26.1 19.9 24.1

37.8 129.5 134.0 172.8 107.8

c

(GPa)

Sm (R)

SV (R)

SD (R)

0.221 1.203 1.047 0.975 0.989

0.092 0.259 0.222 0.378 0.337

0.129 0.944 0.825 0.597 0.652

a

a

Poirier (1991). Fei (1995). c Bass (1995). b

with the R ln 2 rule, while SD is 0 or negative for LiX. The values of SD for NaX lie between the R ln 2 and 0 line (Fig. 14). It is not surprising if we consider the possible large differences in disordering upon melting. Instead of a strict R ln 2 rule, we propose that Sm = xR ln 2 + αKVm

(14)

where 0 ≤ x ≤ 1, although by eliminating large uncertainties in calculating SV (e.g., NaX), data 11

4.2. High-pressure melting parameters based on the Lindemann law and R ln 2 rule

V Sound speed (km/s)

points may cluster near the x = 1 and 0 lines. The structure-dependence of x, should be investigated in a fundamental way and is an important issue for future investigation. For the materials investigated in shock melting experiments (essentially, melting of high-pressure phases of Fe, V, KBr, CsBr, NaCl and silica), we assume that the R ln 2 rule applies (x = 1) as there is no apparent contradiction.

solid

10

Now we deduce the parameters of high-pressure melting which are implied by the previously discussed shock temperature, and sound-speed data for elements, alkali halides and minerals. Eq. (13) can be written alternatively as

9 liquid

8 Dai et al., 2001

7

Sm = R ln 2 + γCV

10000 Dai et al., 2001

T (K)

c' solid

6000

c

Lindemann MC

4000

Vm =

DAC MC extrapolation (Errandonea et al., 2001)

2000 100

150

200

250

(15)

where γ is the Grüneisen parameter and CV the specific heat per mole atom. Both Eqs. (13) and (15) will be used for evaluating Sm with either γCV or αK. Sm is related to Vm via the Clausius–Clapeyron equation

liquid

8000

Vm V

300

P (GPa) Fig. 11. Shock-state sound-speed and temperature measurements for V. DAC melting data of V (Errandonea et al., 2001) is extrapolated to higher pressures.

dTm Sm dP

(16)

The slope dTm /dP along melting curve can be calculated using the Lindemann law (Stacey and Irvine, 1977) d ln Tm 2γ = dP K

(17)

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S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386 11

12000 Mo & W 10000

io

io

on

on

ug

(H

ug

9 melt?

W

o

(H

6000 M

T (K)

t)

t)

8000

8

W (DAC)

4000

Mo (DAC)

Sound Speed (km/s)

10

Mo (sound speed)

7

2000 0

100

0

300

200

6 500

400

P (GPa) Fig. 12. Shock-state sound-speed measurements for Mo (Hixson et al., 1989) and calculated P − T Hugoniots for Mo and W (Hixson and Fritz, 1992). DAC melting data of Mo and W (Errandonea et al., 2001) are extrapolated to higher pressures.

Since in most of the materials of interest (Table 3), melting of a high-pressure phase occurs, we require the Grüneisen parameter and equation of state (EOS) of the high-pressure phase. We thus employ in most cases the bulk modulus that is derived from the principal Hugoniot centered at the zero-pressure density of the metastable high-pressure phase, and assume a Murnaghan EOS (Poirier, 1991). The shock wave EOS, can be described by a linear shock velocity (US )–particle velocity (up ) relation US = C0 + sup

(18)

where C0 and s are the parameters. The bulk modulus is K0 = ρ0 C02 , and its pressure derivative is K0 = 4s − 1. The subscript 0 denotes ambient condition. The Grüneisen parameter, γ, is assumed to be volume dependent only
q V γ(V) = γ0 (19) V0 where γ0 = (162s2 − 360s + 215)/18s (Jeanloz and Grover, 1987). Given a reference state (Pc , Tc ) resolved from shock melting experiments (Table 3), 7.5

12000

Ta 10000

7.0

T (K)

8000

solid

6.5 6000 6.0 4000 DAC (Errandonea et al., 2001)

5.5

2000 0 0

Sound speed (km/s)

liquid

Sound speed measurement Calculated bounds of T

100

200

300

400

5.0 500

P (GPa) Fig. 13. Shock-state sound-speed measurements for Ta, and the upper and lower bound of calculated shock temperature (Shaner et al., 1984) near the melting pressure.

S.-N. Luo, T.J. Ahrens / Physics of the Earth and Planetary Interiors 143–144 (2004) 369–386

383

Table 6 The R ln 2 rule: entropy of melting for alkali halides (heat capacity and entropy refer to per mole atoms)

LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI a b

γ0 a (cm3 /g)

Vm a (cm3 /g)

Vm /V (cm3 /g)

CV (R)b

Sm (R)a

SV (R)

SD (R)

1.63 1.81 1.94 2.19 1.51 1.62 1.65 1.71 1.52 1.49 1.50 1.49

3.47 5.63 6.22 7.22 4.75 7.01 7.67 8.82 5.04 7.02 7.72 8.86

0.356 0.277 0.248 0.222 0.320 0.259 0.239 0.216 0.219 0.187 0.178 0.167

2.52 2.89 2.94 3.01 2.82 3.04 3.09 3.14 2.95 3.09 3.15 3.18

1.451 1.356 1.290 1.187 1.592 1.567 1.539 1.519 1.501 1.529 1.524 1.514

1.458 1.449 1.417 1.465 1.361 1.277 1.220 1.161 0.981 0.860 0.843 0.789

−0.007 −0.093 −0.127 −0.278 0.231 0.290 0.319 0.358 0.520 0.669 0.681 0.725

Jackson (1977). JANAF (Malcolm, 1998).

the melting curve can be calculated by integrating Eq. (17). The Lindemann melting curves are plotted in Figs. 5–11. Sm and Vm at certain pressure, e.g., Pc can be readily calculated as shown in Table 3. Note that the volume change (Vm /V ) upon melting at high-pressure, varies between 0.9 and 1.8% for the materials investigated, thus Sm is close to R ln 2 due to the small changes in volume. The Lindemann melting curve for Fe suggests that Fe melts at ∼6277 K and 330 GPa.

metals silicates alkali halides

x=1 x=0

∆Sm (R)

2

NaX

KX

LiX

1 ln2 quartz

0

0

0.5

1.0

1.5

2.0

∆SV (R)

Fig. 14. The R ln 2 rule of entropy of melting: Sm vs. SV for metals, silicates and alkali halides. Also refer to Tables 4–6. Solid lines have slopes of 1 and intercepts of R ln 2 (x = 1) and 0 (x = 0), respectively. X denotes F, Cl, Br and I.

The Lindemann melting curves constructed via the above procedures demonstrate self-consistency and good agreement with the DAC experiments using laser-heating techniques. The two Lindemann curves for stishovite obtained from fused and crystal quartz agree closely (within 150 K). The Lindemann curve for fused quartz also agrees with the DAC data (Shen and Lazor, 1995) at 25–40 GPa. There is excellent agreement between DAC experiments (Boehler et al., 1996, 1997) and Lindemann curves from shock wave results for KBr and NaCl. The above comparison illustrates the encouraging convergence between the static and dynamic data for silica and alkali halides. But significant discrepancies exist for transition metals. Shock melting pressures for Fe, V, Mo, W and Ta are significantly higher than those achieved in DAC melting experiments, thus a comparison requires large extrapolation of the DAC melting data. If we extrapolate the DAC melting curves (Errandonea et al., 2001) for Fe, V, Mo, W and Ta to 200–400 GPa where shock melting occurs, shock temperature measurement and calculation would indi+ cate ΘH ∼ 0.7–2.0 (Figs. 9–13). These large values of superheating are not consistent with the superheating systematics, even when the uncertainties in shock temperature and β variations at high pressures are taken into account. The discrepancies could be reconciled by possible solid–solid phase transitions at high pressures which may steepen the melting curves at higher pressures, and bring the convergence of static and dynamic experimental results. In particular, Fe could

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undergo a solid–solid phase transition at ∼200 GPa (Brown, 2001) and melt at ∼270 GPa upon shock wave loading.

5. Conclusions Superheating systematics β = (A0 −b log Q)θc (θc − 1)2 based on undercooling experiments incorporate heating rates and predict the tightest upper bound among a host of methods reported to date. For Q ∼ 1012 K/s, the maximum superheating predicted for crystalline materials, varies between 0.05 and 0.43Tm . We examined the shock superheating-melting data on silicates, alkali halides and metals against the superheating systematics, and validate the method of resolving the equilibrium melting point (Pc , Tc ) on Hugoniots. The agreement between shock-induced superheating and the superheating systematics indicates that the assumption of homogeneous nucleation of melt upon shock wave loading of single crystals is viable. We parameterize the R ln 2 rule for disordering entropy of melting as SD = xR ln 2. x = 0 for quartz and lithium halides at ambient pressure. x is taken to be 1 for the high-pressure phases investigated. With (Pc , Tc ) resolved from shock wave experiments, the melting curves at high pressures, entropy and volume changes upon melting were obtained for stishovite, KBr, CsBr, NaCl (B2 phases), Fe and V based on the Lindemann law and R ln 2 rule. The volume change (Vm /V ) upon melting at Pc , varies between 0.9 and 1.8% for the materials investigated, thus Sm is close to R ln 2 due to the small changes in volume. For alkali halides and silica, the values for (Pc , Tc ) and corresponding Lindemann melting curves, are in excellent agreement with the DAC results. For transition metals, significant discrepancies exist between shock wave results and the extrapolation of DAC data. Simple extrapolations of the DAC melting curves to the pressure range of shock melting indicate super+ heating of ΘH = 0.7–2.0 for Fe, V, Mo, W and Ta. These are far beyond the predictions of superheating systematics, even if we consider the uncertainties in shock temperature measurement (or estimation) and pressure-induced changes in β. Possible high-pressure solid–solid phase change—thus a change in slope, dTm /dP—could reconcile these discrepancies, al-

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